Hann function: Difference between revisions

The '''Hann function''' of length <math>L,</math> used to perform '''Hann smoothing''',<ref>{{Cite book|url=http://worldcat.org/oclc/152410575|title=Elements of statistical analysis|last=Essenwanger, O. M. (Oskar M.)|date=1986|publisher=Elsevier|isbn=0444424261|oclc=152410575}}</ref> is named after the Austrian meteorologist [[Julius von Hann]], is a [[window function]] given by''':'''

where the length of the window is <math>N+1,</math> and N can be even or odd. (see [[Window function#Hann and Hamming windows]]) It is also known as the '''raised cosine window''', '''Hann filter''', '''von Hann window''', etc.<ref name=":1">{{Citation|last=Kahlig|first=Peter|titlechapter=Some aspects of Julius von Hann's contribution to modern climatology|date=1993|chapter-url=https://www.researchgate.net/publication/260824978_Some_aspects_of_Julius_von_Hann%27s_contribution_to_modern_climatology|work=Geophysical Monograph Series260824978|volume=75|pages=1–7|editor-last=McBean|editor-first=G.A.|publisher=American Geophysical Union|language=en|doi=10.1029/gm075p0001|isbn=9780875904665|quote=Hann appears to be the inventor of a certain data smoothing procedure, now called "hanning" ... or "Hann smoothing" ... Essentially, it is a three-term moving average (running mean) with unequal weights (1/4, 1/2, 1/4).|access-date=2019-07-01|editor2-last=Hantel|editor2-first=M.|title=Interactions Between Global Climate Subsystems: The Legacy of Hann|series=Geophysical Monograph Series}}</ref><ref>{{Cite book|url=https://ccrma.stanford.edu/~jos/sasp/Hann_Hanning_Raised_Cosine.html|title=Spectral audio signal processing|last=Smith, Julius O. (Julius Orion)|first=|date=2011|publisher=W3K|others=Stanford University. Center for Computer Research in Music and Acoustics., Stanford University. Department of Music.|year=|isbn=9780974560731|location=[Stanford, Calif.?]|pages=|oclc=776892709}}</ref>

The function is named in honour of von Hann, who used the three-term weighted average smoothing technique on meteorological data.<ref>{{Cite book|url=https://books.google.com/books?id=Ma3PAAAAMAAJ&pg=PA199|title=Handbook of Climatology|last=Hann|first=Julius von|date=1903|publisher=Macmillan|year=|isbn=|location=|pages=|language=en|quote=The figures under ''b'' are determined by taking into account the parallels 5° away on either side. Thus, for example, for latitude 60° we have ½[60+(65+55)÷2].}}</ref><ref name=":1" /> However, the erroneous<ref name=":0" /> "Hanning" function is also heard of on occasion, derived from the paper in which it was named, where the term "hanning a signal" was used to mean applying the Hann window to it.<ref>{{Cite journal|last=Blackman|first=R. B.|last2=Tukey|first2=J. W.|date=1958|title=The measurement of power spectra from the point of view of communications engineering — Part I|url=https://ieeexplore.ieee.org/document/6768513/|journal=The Bell System Technical Journal|volume=37|issue=1|pages=273|doi=10.1002/j.1538-7305.1958.tb03874.x|issn=0005-8580|via=}}</ref><ref>{{Cite book|url=https://archive.org/details/TheMeasurementOfPowerSpectra/page/n58|title=The measurement of power spectra from the point of view of communications engineering|last=Blackman|first=R. B. (Ralph Beebe)|last2=Tukey|first2=John W. (John Wilder)|date=1959|publisher=New York : Dover Publications|year=|isbn=|location=|pages=98|lccn=59-10185}}</ref> The confusion arose from the similar [[Hamming function]], named after [[Richard Hamming]].